The  Poles  of  a  Right  Line 


WITH 


RESPECT  TO  A  CURVE  OF  ORDER  n 


A  THESIS 


Pbesented  to  the  Faculty  of  Philosophy  of  the 
University  of  i  Pennsylvania 


jl'ENNS 


By 


ROXANA  HAYWARD  VIVIAN 


In  Partial  Fulfilment  of  the  Requirements 

fob  the  degree 

Doctor  of  Philosophy 


PHILADELPHIA 
1901 


The  Poles  of  a  Right  Line 


WITH 


RESPECT  TO  A  CURVE  OF  ORDER  fl 


A  THESIS 

Peesented  to  the  Faculty  of  Philosophy  of  the 
University  of  Pennsylvania 


By 
EOXANA   HAYWARD   VIVIAN 


In  Pabtial  Fulfilment  of  the  Requirements 

for  the  degree 

Doctor  of  Philosophy 


TOi 


PHILADELPHIA 
1901 


•  •    •   • 


V5- 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY, 

LANCASTER,   PA. 


THE  POLES  OF  A  EIGHT   LINE  WITH   RESPECT 
TO   A   CURVE   OF   ORDER   n. 

The  short  article,  "  Allgemeine  Eigenschaften  der  Algebra- 
ischen  Curven,"  published  by  Steiner  in  Crelle's  Journal,  vol. 
XLVII,  pp.  1-6,  has  been  the  starting  point  for  many  investiga- 
tions in  the  theory  of  polar  curves  and  envelopes.  His  theorems, 
stated  without  proof,  are  given  with  reference  to  the  general 
curve  of  order  n .  Salmon  in  "  Higher  Plane  Curves,''  ^  gives 
a  r^sum^  of  Stein er's  theorems  with  reference  to  the  general 
curve,  and  a  more  specific  discussion  of  the  cubic.  He  uses  a 
method  partly  analytic  and  partly  geometric.  The  most  com- 
plete treatment  of  the  subject  of  poles  and  polars  is  that  of 
Cremona  in  his  "  Introduction  to  the  Study  of  Plane  Curves," 
published  in  1865.  He  bases  the  theory  entirely  upon  the 
properties  of  loci  of  harmonic  means ;  and  his  purpose,  stated 
in  the  preface,  is  to  give  a  satisfactory  geometrical  proof  of  the 
theorems  enunciated  by  Steiner  and  other  writers  on  the  subject. 
These  three,  Salmon,  Cremona,  and  Steiner,  have  considered  the 
question  of  poles  and  polars  from  the  standpoint  of  the  general 
curve  of  order  n ,  and  with  important  articles  by  Clebsch  and 
others,  to  which  references  will  be  made  later,  constitute  the 
chief  sources  for  the  general  theory  of  polar  curves. 

By  no  one  of  these  writers  has  a  detailed  study  been  made  of 
the  poles  of  a  right  line,  or  the  (n  —  1)^  intersections  of  the  first 
polar  curves  of  points  in  a  right  line,  with  respect  to  a  curve  of 
order  n .  Steiner  in  general  considers  such  points  as  the  envelope 
of  first  polar  curves  of  points  on  a  given  locus.  In  case  the 
directrix  is  a  right  line,  the  envelope,  by  his  formula,  reduces  to 
order  zero.  Cremona  studies  them  from  the  standpoint  of  base 
points  of  a  pencil  of  curves  of  order  n  —  1 ,  while  both  Salmon 
and  Cremona  call  them  specifically  "poles''  of  the  line,  and 
give  some  limitations  to  their  position  in  the  case  of  cubics. 

iPp.  357-368. 


594330 


4  THE  POLES   OP  A  RIGHT  LINE 

It  is  proposed  in  this  paper  to  investigate,  by  analysis  as  far 
as  possible,  the  character  and  position  of  the  pbles  of  a  right 
line  L  =  $x  -{-  7^y  -}-  ^z  =  0  in  the  different  relations  it  may  have 
with  respect  to  a  base  curve  Z7=  0,  whose  equation  is  homo- 
geneous and  of  order  n,  and  certain  curves  derived  from  U, 
The  cases  for  any  of  these  loci  of  singularities  of  higher  order 
than  a  triple  point  formed  by  three  ordinary  branches,  or  a  cusp 
with  an  ordinary  branch  passing  through  it,  will  not  be  consid- 
ered except  in  discussing  the  relation  of  tangents  to  first  polar 
curves  at  certain  points  of  higher  order  of  multiplicity. 

§1. 

The  Pencil  of  Curves  of  Which  the  Poles  are  Base 
X  Points. 

The  system  of  first  polar  curves  with  respect  to  Z7=  0  is  of 
the  form  xU^  -\-  ylJ^  -\-  zU^^  0 ,  where  (a; ,  i/ ,  »)  is  kny  point  in 
the  plane.  Taking  the  polars  of  all  points  in  a  line  determined 
by  two  fixed  points,  the  system  reduces  to  a  pencil  of  curves 
projectively  related  to  points  on  that  line,  and  their  (n  —  1)^  in- 
tersections are  the  poles  of  the  line.  These  curves  determine 
on  the  line  an  involution  of  degree  n  —  1 ,  and  the  2(n  —  2) 
double  points  of  this  involution  are  the  points  where  curves  of 
the  pencil  touch  the  line.^  Any  curve  of  the  pencil  is  com- 
pletely determined  when  one  point  in  addition  to  the  {n  —  ly 
base  points  is  known.  If  this  point  is  taken  infinitely  near  a 
base  point  it  determines  there  the  tangent  to  a  single  curve  of 
the  pencil,  and  the  pencil  of  curves  and  the  pencil  of  tangents 
are  projectively  related  and  have  a  (1 ,  1)  correspondence  with 
each  other  and  with  the  points  of  the  line.  When  two  of  the 
polars  touch,  the  two  pencils  of  tangents  belonging  to  the  two 
coinciding  base  points  reduce  to  a  common  tangent  to  all  the 
curves  except  the  one  which  has  there  a  double  point ;  if  a  base 
point  is  a  multiple  point  of  order  r  for  all  curves  of  the  pencil 
and  these  have  r  common  tangents  at  the  point,  one  curve  of  the 

1  Clebsch,  Crdle,  vol.  LVIII,  p.  280. 


WITH   RESPECrr   TO   A   CURVE   OF   ORDER  n.  5 

pencil  has  the  base  point  for  a  multiple  point  of  order  r  +  1 ; 
and  in  general  all  the  properties  which  hold  for  a  pencil  of 
curves  will  be  true  for  these.  ^ 

The  coordinates  of  the  poles  of  ^a;  +  ^y^/  +  C^  =  0 ,  or  the  base 
points  of  the  pencil  of  first  polars  of  the  points  in  the  line,  are 
given  by  the  intersections  of 

§2. 
The  Kelated  Curves. 

Closely  connected  with  the  theory  of  polar  curves  are  the 
Hessian  and  the  Steinerian  of  the  base  curve,  and  through  them 
the  Cayleyan.  In  general,  that  is  when  U  is  non-singular,  they 
are  of  orders  3(7i  —  2)  ,  3(n  —  2)^,  and  3(n  —  2)  (5yi  —  1 1)  respec- 
tively, and  the  Hessian  has  no  double  points.^  The  Steinerian 
and  the  Cayleyan  have  a  (1 ,  1)  correspondence  with  the  Hes- 
sian. In  addition  to  their  ordinary  definitions  it  will  be  con- 
venient to  characterize  the  Hessian  as  the  locus  of  coincident,  or 
double,  poles  of  a  line,  and  the  Steinerian  as  the  envelope  of 
line  polars  of  points  on  the  Hessian.^  The  corresponding  points 
on  the  two  curves  are  in  the  same  relation  as  those  which  Pro- 
fessor Cay  ley  calls  "  conjugate  poles  "  for  the  cubic,''  and  the 
Cayleyan  is  the  envelope  of  lines  joining  conjugate  poles.  There 
is  also  another  locus  upon  which  lie  all  the  inflexions  of  first 
polar  curves  for  the  pencil  belonging  to  X  =  0,  and  which  has 
the  base  points  of  the  pencil  for  triple  points.  The  equation 
and  certain  important  characteristics  of  this  curve  will  be  de- 
veloped later. 

The  equations  of  the  Hessian  and  the  Steinerian  are  deduced 
from  the  condition  that  any  polar  may  have  a  double  point. 

^  Cremona,  Introdiwtion,  §  10. 

2  Salmon,  Higher  Plane  Curves,  p.  363. 

3  Cremona,  §  20,  No.  118a. 

*  Memoir  on  Chirves  of  the  Third  Order,  Collected  Papers,  vol.  II,  p.  382. 


6 


THE  POLES  OF  A  RIGHT  LINE 


Let  Fj  =  iTj  ZJj  +  3/1  ZJg  +  2j  ZJg  =  0  be  the  polar  of  the  point 
(jCj,  2/j,  2j)  on  i  =  0 .  If  this  has  a  double  point  at  (x,  y' ,  z)y 
we  have  the  identical  relations 


J*  F,  =  (a,a;  +  a^  +  v)  (A*  +  /^^^  +  ^i') . 

«x  ^ii2  +  yi  c^;22  +  «i  ^m = «A  +  «A » 

«1  ^(23  +  Vl  ^223  +  «1  ^^  =  «2^3  +'^Ay       . 
«1  f^l'33  +  Vl  ^233  +  ^1  ^333  =  2«3^3  • 


0, 


Eliminate  the  a's ,  ^'s ,  and  x^,  y^,  z^,  and  the  Hessian  is  ob- 
tained 


=  0, 


V'n 

U'u 

V'n 

U'n 

U',, 

U'^s 

U[, 

V'^ 

U'^s 

or  the  locus  of  double  points  on  first  polar  curves.  The  Stein- 
erian  may  be  found  by  eliminating  the  a's ,  /3's  and  x'  ^  y\  z  \ 
although  to  express  it  in  algebraic  form  is  practically  impossible 
for  curves  of  higher  order  than  the  fourth.  The  intersections 
of  li  with  the  Hessian  and  the  Steinerian  give  respectively  the 
number  of  double  points  on  first  polar  curves  which  lie  on  X,  and 
the  number  of  points  whose  first  polars  have  a  double  point  for 
the  pencil  of  curves  belonging  to  X . 

All  points  which  may  be  cusps  on  first  polar  curves  lie  on  the 
Hessian,  and  also  on  a  curve  of  order  4(71  —  3),  which  is  ob- 
tained by  using  the  condition  a  =  ^  and  eliminating  a  and x^^y^^ 


WITH  EESPECT  TO  A  CURVE  OF  ORDER  n.         7 

z^ .     The  corresponding  points  (x' ,  y\  z')  are  cusps  on  the  Stein- 
erian,  and  are  12(ri  —  2){n  —  3)  in  number.^ 

The  first  polars  V^  and  V^  of  two  points  which  determine  a  line 
will  in  general  have  no  common  intersections  which  lie  on  either 
X,  TJ,  or  the  curves  just  mentioned;  and  poles  which  do  not 
lie  on  i,  TJ  y  or  the  Hessian  may  be  conveniently  termed 
"  free  "  poles.^  Poles  which  lie  on  the  Steinerian  and  the  Cay- 
leyan  are  included  among  the  free  poles,  since  poles  on  these  loci 
need  not  satisfy  conditions  of  the  particular  kind  which  govern 
the  others.  Any  pole  of  a  line  will  be  a  triple  point  of  the 
general  inflexion  locus,  which  is  not  fixed  as  are  the  Hessian, 
Steinerian,  and  Cayleyan,  by  the  base  curve,  but  varies  with  the 
line.  The  number  of  free  poles  will  be  diminished  and  their 
characteristics  will  be  changed  as  the  line  is  defined  by  special 
relations  to  U  and  the  Steinerian,  or  as  singularities  are  intro- 
duced into  the  base  curve.  In  no  case,  however,  can  the  num- 
ber of  fi'ee  poles,  depending  only  on  the  base  curve,  be  less  than 
n—\  while  C/" is  a  proper  curve  with  none  of  the  complex  singu- 
larities ;  for  at  the  multiple  point  of  highest  order,  n  •—  1 ,  the 
first  polars  have  (n  —  2)^  intersections,  and  these,  with  n  —  2 
additional  common  points  if  all  the  branches  are  cuspidal,  leave 
n  —  \  free  poles. 

§3. 

Poles  when  U  has  no  Double  Points  or  Other 
Singularities. 

Under  this  hypothesis  the  Hessian  has  no  singularity. 

I.  A  line  which  has  only  ordinary  intersections  with  TJ  and 
the  Steinerian  can  have  only  free  poles,  as  is  evident  from  the 
conditions  which  exist  when  a  pole  lies  upon  either  of  these  loci. 

^  Clebsch,  Z7e5er  Qwrvm,  vierter  Ordnung,  CreUe,  Vol.  LIX,  p.  130.  Cf.  also, 
Steiner's  article  in  Crdle,  Vol.  XLVII,  pp.  1-6,  andHenrici's,  Proc.  Lon.  Math. 
Soc.,  Vol.  II,  p.  112. 

2  Salmon  in  his  discussion  of  the  cubic  omits  from  the  number  of  poles  those 
which  occur  at  a  double  point  or  cusp  ;  but  it  seems  better  to  include  these  in 
the  total  number,  since  they  come  under  the  definition,  and  have  all  the  charac- 
teristics of  poles. 


8  THE  POLES  OF   A   RIGHT  LINE 

(a)  If  a  pole  lies  on  L  it  must  be  either  a  point  of  tangency  of 
the  line  with  11,  or  a  double  point  on  U.  j 

Let  (x'y  y'f  z),  a  pole  of  the  line,  satisfy  the  equation  i  =  0. 
The  coordinates  of  any  point  in  i  =  0  satisfy  the  relation      ^ 

(1)  xU[+yU',+zU',  =  0,  I 

including  those  of  the  pole  itself,  so  that 

hence  {x\  y\  z)  is  on  the  curve  Z7.  For  (1)  to  hold  for  every 
point  in  L  the  line  must  be  tangent  to  CT  at  {x' ,  y  ,  z) ,  ov  else 
U[j  U'^y  U'^  must  vanish  identically  and  (x,  y',  z')  is /a  double 
point  on  U,  Thus  L  must  have  two  points  in  common  with  U 
at  (Xy  y' y  z)y  whlch  may  be  consecutive  or  coincident,  the  two 
conditions  having  the  same  value  here.^ 

(6)  If  a  pole  lies  on  C/"  by  a  similar  method  it  can  be  shown 
to  be  a  point  of  tangency  of  L  with  the  curve,  or  else  a  double 
point  on  the  curve. 

(c)  If  a  pole  lies  on  the  Hessian  L  is  tangent  to  the  Stein- 
erian  at  the  corresponding  point. 

Let  {x  y  y'y  z)  be  a  pole  of  the  line  and  a  point  in  the  Hes- 
sian. It  is  then  a  double  point  on  a  first  polar  curve,  for  ex- 
ample, the  first  polar  F^  of  (x^^ ,  y^ ,  z^)  which  is  an  intersection 
of  L  and  the  Stei    nian.     Now 

V,  =  x,U,-hy,U,-\-z^U,=  0; 

and  since  it  has  a  double  point  at  (a?',  3/',  2') 

^^U[,  +  y,U'^  +  z,U',,=  0,  1 

1  The  two  conditions  are  equivalent  algebraically  and  geometrically  here, 
though  it  may  happen,  as  in  the  case  when  the  line  passes  through  a  double 
point  on  the  Steinerian,  that  they  are  only  equivalent  algebraically.  Cf.  Jon- 
quiSres,  Sur  les  problemes  de  contact  des  courbes  cdgebriques,  Orelle,  Vol.  LXVI, 
p.  291. 


WITH   KESPECT   TO   A   CURVE   OF   ORDER   n.  9 

If  (a?2 ,  ^2  J  ^2)  ^^  ^  second  point  on  L ,  any  other  point  on  the 
line  has  coordinates  (x^  +  Xx^,  y^  +  Xy^,  z^^-\-  Xz^  ;  and  examining 
the  tangent  at  (ic',  y',  z)  to  the  first  polar  of  any  point  {x,  3/,  z) , 
we  have 

xU[,-hy  U[,  +  ^ U[,  =  A(a^^ C7;^  +  y^ U[,  +  ^^^ ^^{3)  > 

^U[,  +  y  U',,  +  zU'^^^  X(x^ U[^  +  y^ ^22  +  h ^23)  > 

a;C7;3  +  y  i7;3  +  zU',,  =  ^O'c^^Ia  +  2/2^^3  +  ^2^;3) ; 

hence  the  tangents  to  all  first  polars  of  the  pencil  at  {x' ,  y' ,  z') 
coincide  with 

+  4^,  c^;,  +  y,  t^;3  +  2,  zz;,]  =  0 . 

A  pole  on  the  Hessian  is  therefore  in  general  composed  of  two 
coincident  poles.  ^  Such  a  pole  corresponds  also  to  a  double 
point  on  the  polar  envelope  of  a  curve,  but  for  a  directrix  of  the 
first  degree  the  envelope  consists  simply  of  the  {n—  Vf  poles 
of  the  line  and  is  of  order  zero.  The  consecutive  curve  to  V^ 
will  have  common  tangents  with  it  and  therefore  a  double  point 
at  {x\  y' ,  2').  The  consecutive  point  on  L  is  on  the  Steinerian 
and  the  line  is  tangent  to  the  Steinerian  at  the  point  {x^ ,  y^ ,  z^ 
which  corresponds  to  {x' ,  y\  z)  ?  It  might  seem  as  in  a  pre- 
ceding case  that  a  line  through  a  double  point  on  the  Steinerian 
would  satisfy  the  same  condition,  but  it  will  be  shown  later  that 
a  pole  on  the  Hessian  will  result  only  in  a  special  case.  The 
above  analysis  is  in  agreement  with  Cremona's  theorems  ^ :  If  two 
curves  of  a  pencil  touch  the  point  of  tangency  counts  for  two 
double  points  and  to  it  corresponds  the  point  where  the  line 
touches  the  Steinerian. 

At  any  pole,  as  has  been  stated,  the  tangents  to  first  polars 
form  a  pencil  of  lines,  and  when  two  curves  of  the  pencil  touch 
it  is  geometrically  evident  that  the  two  pencils  belonging  to  the 

1  These  may  be  called  poles  of  order  2. 

*  The  tangents  to  the  Steinerian  are  thus  line  polars  of  points  on  the  Hessian. 

^Introduction,  §14,  No.  88a,  and  §  19,  No.  112a. 


10 


THE   POLES   OF   A   RIGHT   LINE 


two  coincident  poles  reduce  to  a  single  line,  the  common  tangent. 
The  condition  for  coincident  poles  is  that  the  "pole^  lie  on  the  Hes- 
sian, and  it  is  interesting  to  see  that  the  analytic  condition  for 
the  reduction  of  the  pencil  is  the  same.  Using  a  method  sim- 
ilar to  that  used  by  Clebsch,^  L  may  be  defined  by  two  points, 
{x^,  2/j,  zjand  {x^,  y,^,  z^,  such  that  the  tangents  to  their  respec- 
tive polars  at  the  pole  {x' ,  y\  z)  cut  L  in  those  points.  For  let 
the  line  through  (x^,  Vi*^^  ''^^^  {^'  y  y\  2')  be  «ic  +  i^y  -|-  j'z  =  0 ; 
then 

^'  +  ?y'  +  p'  -  0, 

«^i  +  /5yi  +  r^i  =  0 ; 

and  if  this  line  coincides  with  the  tangent  to  the  polar  of 
(^u  2/u  ^1)     at     {x\  y,  z') 

^1  ^12  +  Vl  ^22  +  \  ^23  =  ^?^ 
^1^13  +  2/l^23  +  ^1^33  =  ^r; 

hence  eliminating  x^ ,  y^  and  z^ , 


=  0. 


There  are  then  two  lines  (a,  ^ ,-f)  through  (a;',  y  ,  z)  which 
contain  points  whose  first  polars  touch  them  at  (x' ,  y\  z).  It 
is  also  evident  that  there  are  2(7i  —  2)  points  in  any  line  where 
first  polars  of  points  in  the  line  may  touch  it. 

X  =  0  may  then  be  defined  by  the  pair  of  points  (x^j  y^j  z J 
and  (rTg,  y^,  z^)  in  the  two  lines  through  (x' ,  y' ,  z)  whose  re- 
spective polars  touch  the  lines  there  ;  and  the  condition  that  the 
lines  coincide  is 


^n 

^n 

U'n 

a 

^u 

U'n 

U',, 

/? 

U[, 

U',^ 

Ul^ 

r 

a 

? 

r 

0 

»  Orelk,  Vol.  LVIII,  pp.  279-280.    Cf.  also  Salmon,  Higher  Plane  Curves,  pp. 
343-344.  • 


U'n 

U[, 

^n 

U[, 

m. 

U',^ 

U[, 

^n 

c^;a 

WfTH   EESPECT  TO  A  CURVE   OF   OEDEE  Jl.  11 


=  0, 


SO  that  {x' y  y\  z)  is  on  the  Hessian. 

These  pairs  of  lines  are  what  Cremona  calls  the  "  indicatrices  " 
of  a  point — the  pair  of  tangents  which  can  be  drawn  from  it  to 
its  conic  polar.^ 

(d)  If  a  pole  lies  on  the  Steinerian  it  is  a  double  point  on  a 
polar  conic.  This  is  simply  the  definition  of  the  Steinerian, 
and  is  all  that  can  be  affirmed  in  regard  to  such  a  pole ;  for  the 
reciprocal  relation  between  the  Hessian  and  the  Steinerian  does 
not  lead  to  the  reciprocal  characteristic  that  the  tangents  to  the 
Hessian  are  line  polars  of  points  on  the  Steinerian,  except  in 
the  case  of  cubics  when  the  two  loci  coincide.  Corresponding 
to  the  points  on  L,  there  is  a  pencil  of  first  polars,  V^-^  XV^y 
which  represents  all  the  first  polar  curves  of  points  on  the  line, 
and  these  all  pass  through  the  {n  —  ly  poles.  The  case  is  dif- 
ferent for  the  polar  conies  of  points  on  a  line,  which  are  of  the 
form 

where  U^^y  '^ 2i'>  ^^^v  ^^®  ^^  degree  n  —  2  in  the  coordinates  of 
the  point  and  the  system  is  not  a  pencil.  The  conies  do  not  all 
pass  through  the  pole  on  the  Steinerian  and  the  preceding  analy- 
sis cannot  be  applied. 

If  the  line  is  tangent  to  the  Hessian  at  (x'j,  y' ,  z)  the  polar 
conies  of  the  two  intersections  have  double  points  at  the  corre- 
sponding point  pn  the  Steinerian.  If  this  is  a  pole  of  the  line  it 
is  not  to  be  distinguished  from  the  case  where  the  line  simply  in- 
tersects the  Hessian,  for  the  number  of  polar  conies  which  may 
have  a  double  point  at  the  pole  is  not  taken  into  account. 

A  pole  on  the  Steinerian,  therefore,  is  the  result  of  the  per- 
fectly general  condition  that  the  line  considered  is  the  line  polar 

^  Cremona,  Introduction,  §14,  No.  90c,  and  |19,  No.  112.  The  Hessian  with 
the  base  curve  forms  the  locus  of  points  for  which  the  indicatrices  reduce  to  a 
single  line. 


12  THE   POLES    OF   A   RIGHT   LINE 

of  the  point,  and  it  is  numbered  among  the  free   poles.     The 
same  is  true  for  the  Cayleyan. 

II.  It  follows  immediately  from  I  that  when  L  has  points  of 
tangeucy  with  TJ  and  the  Steinerian  the  number  of  free  poles  is 
diminished. 

If  L  has  only  ordinary  points  of  contact  with  TJ  each  point  of 
tangency  is  a  pole  and  lies  on  both  the  loci.  The  case  of  asymp- 
totic tangents  is  a  special  case  which  gives  a  pole  at  infinity. 
The  maximum  number  of  such  poles  which  may  lie  on  U  is 
[n/2] ,  leaving  {2n  —  1)  (?i  —  2)/2  possible  free  poles  when  n  is 
even,  and  (n  —  1)  (2n  —  3)/2  when  n  is  odd. 

If  L  is  tangent  to  U  at  an  inflexion,  it  is  tangent  to  the  Stein- 
erian as  well,  and  has  a  pole  on  the  Hessian,  the  point  of  inflex- 
ion ;  for  all  first  polars  of  its  points  touch  the  line  at  the  point  of 
inflexion.  Such  a  pole  is  of  order  2  and  is  on  the  three  loci,  L , 
Uy  and  the  Hessian.  The  number  of  double  poles  which  may 
arise  from  inflexional  tangency  cannot  be  greater  than  [n/3] . 

If  L  is  tangent  to  the  Steinerian,  but  not  an  inflexional  tan- 
gent to  U,  there  is  again  a  pole  on  the  Hessian  at  the  corre- 
sponding point  and  necessarily  a  double  pole.  The  maximum 
number  of  double  poles  is  [3(?i  —  2)72] . 

Combinations  of  simple  and  inflexional  tangency  with  C/^give 
corresponding  combinations  of  single  and  double  poles.  A  line 
which  has  more  than  one  simple  point  of  tangency  with  the 
Steinerian  ^  also  presents  no  difficulty  in  classifying  the  poles 
fixed  by  the  given  conditions.  Combinations  of  tangency,  how- 
ever, between  U  and  the  Steinerian,  depend  upon  the  number  of 
conditions  L  can  satisfy  with  respect  to  these  curves.  If  a 
method  could  be  devised  for  finding  the  maximum  number  of 
times  a  line  may  be  tangent  to  these  curves  simultaneously,  the 
maximum  number  and  character  of  the  poles  defined  by  these 
conditions  would  follow  immediately.  But  the  investigation  for 
a  special  and  simple  case  is  rendered  practically  impossible  on 
account  of  the  difficulty  of  obtaining  the  Steinerian  in  a  suitable 
form. 

1  Cremona,  Introduction,  g  20,  No.  119.  If  a  line  is  a  double  tangent  to  the 
Steinerian  all  first  polars  touch  at  the  two  corresponding  points  on  the  Hes- 


WITH   RESPECT   TO   A   CURVE   OF   ORDER   n,  13 

§4. 

The  Inflexion  Locus. 

The  position  of  inflexions  on  polar  curves  does  not  in  any- 
way limit  the  number  of  free  poles  of  a  line,  but  a  consideration 
of  the  inflexions  serves  to  define  somewhat  the  character  of  the 
poles,  and  before  turning  to  the  case  where  TJ  has  double  points 
or  other  singularities  it  will  be  convenient  to  discuss  the  gen- 
eral inflexion  locus,  and  the  inflexion  cubic  for  any  point  in  the 
plane  and  in  particular  for  any  pole  of  the  line  L . 

Referring  to  §2,  if  the  polar  V^  of  (ic^,  y^,  z^)  has  an  inflex- 
ion at  (ic',  y' y  z  )j  the  six  equations  involving  C/j^^,  TJ^^^y  etc., 
must  exist,  and  also 

J2  F,  =  (  «,a;'  +  «^ '  +  a^:  )  (  ^,x  +  /9^  +  /93.  )  =  0 . 

Eliminating  the  a's  and  /3's,  the  point  (a;^,  y^y  2^)  must  satisfy 
the  two  conditions 

%  f^^n  4-  2/1  '^\vi  +  zi  'U\xz      ^\  'U'xvi.  4-  Vx  ^^22  +  ^\  ^^23      H  C^^is  +  Vx  t^^23  +  %  ^^^'33 
a^i  'U\xi.  +  Vx  t^'i22  +  zi  ^^'123     ^  U\^^  +  Vx  U\22  +  2i  U' ^^     »i  ?7^23  +  Vx  U'm  -h  h  U\^^ 

xx  u\ii  +  Vi  u^^z + zi  U'^as    ^1  u\2s  4-  yi  u^22z  +  2i  ^''233    a^i  u\^s + yi  t/'''233  +  h  U'333 

hence  there  are  three  points  whose  first  polars  satisfy  the  given 
condition.  Clebsch  deduces  from  this  the  theorem  :  "  There  are 
always  three  different  poles  whose  polars  have  an  inflexion  at  a 
given  point  '';^  but  an  examination  of  the  conditions  under  which 
the  determinant  cubic  is  derived,  as  well  as  a  study  of  special 
cases,  shows  that,  strictly  speaking,  an  inflexion  will  not  always 
result.  The  locus  evidently  includes  all  points  whose  first 
polars  have  three  consecutive  or  coincident  points  at  (  Xy  y'y  z' ) 
in  the  same  straight  line.  It  will  however  be  convenient  to 
adopt  a  broader  meaning  of  the  term  "  inflexion,'^  as  it  is  used 
by  Clebsch  and  other  writers,  to  include  all  the  cases  which  fol- 

1  Orelle,  Vol.  LIX,  p.  127. 


=  0 


14  THE   POLES   OF   A   EIGHT   LINE 

low,  and  should  be  so  understood  throughout  this  section.  The 
lines  thus  related  to  first  polar  curves  may  be  inflexional  tan- 
gents in  the  ordinary  sense,  tangents  at  a  double  point,  any  line 
through  a  triple  point,  or  a  straight  line  through  the  point  which 
forms  part  of  a  degenerate  polar.  One  point  will  always  have 
corresponding  to  it  a  real  line  fulfilling  the  given  conditions, 
while  the  other  two  lines  may  be  conjugate  imaginaries. 

The  locus  of  all  inflexions  of  polar  curves  of  points  on  L  is  ob- 
tained by  eliminating  (x^ ,  y^ ,  z^  from 

and  the  determinant  cubic  above.  The  resulting  equation  is  of 
degree  6  (n  —  2)  and  has  the  base  points  of  the  pencil  for  triple 
points.  It  is  evident  that  points  on  this  locus  which  are  double 
points  on  polar  curves  are  also  on  the  Hessian,  so  that  the  two 
curves  are  closely  connected. 

When  the  point  (x^ ,  y^ ,  z^  whose  polar  has  an  inflexion  at 
(a;',  y' ,  2')  describes  the  line  X,  there  is  a  third  condition  for  it 
to  satisfy,  and  in  general  the  three  loci  will  have  no  common  in- 
tersection, or  one.     But  at  an  ordinary  pole  of  the  line 

L^^x^-\-  riy^  -h  a^z^  =  X{x^U[  +  y^U'^  +  z^L'^)  =  0, 

and  any  line  has,  corresponding  to  each  pole,  a  set  of  three  points 
whose  first  polars  have  an  inflexion  at  the  pole.  These  are  the 
intersections  of  the  line  with  the  cubic  determinant  belonging  to 
that  pole,  and  the  sets  of  three  are  {n  —  Xf  in  number.  It  fol- 
lows from  this  property  of  the  poles-  that  they  are  triple  points 
of  the  inflexion  locus  of  the  line. 

As  a  special  case  a  tangent  to  Z7=  0  contains  three  points 
whose  first  polars  have  an  inflexion  at  the  point  of  tangency. 

The  term  "  inflexion  cubic  "  for  any  point  will  be  used  to  des- 
ignate the  determinant  cubic  when  the  coordinates  of  the  point 
have  been  substituted  in  U^^^ ,  U^^^ ,  etc.  These  curves,  the  in- 
flexion cubic  and  the  inflexion  locus,  are  both  derived  from  U, 
but  they  are  evidently  not  dependent  upon  U  alone,  as  are  the 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  n.  15 

Hessian,  Steinerian  and  Cayleyan,  since  the  former  varies  for 
every  point  in  the  plane  and  the  latter  with  every  line.  It  is 
necessary  to  examine  the  inflexion  cubic  corresponding  to  a  double 
point  or  cusp  on  the  base  curve. 

If  CT'has  a  double  point  by  a  suitable  choice  of  axes  and  co- 
ordinates its  equation  may  be  written  in  the  form 

+  (a'x^  -f  b'x^y  +  c'xy  +  d'a^y  +  eyy-""  +  .  • .  =  0 , 

where  (0,0,  1)  is  the  double  point  with  tangents  aj  =  0  and 
y  =  0.     Thence  we  obtain 

ZZui  =  Qaz"^^  +  (24a'x  +  Wy)z'^^  +  •  •  • , 

U^,^  =  2hz^-^  +  {Wx  +  4c»"-^  +  . . . , 

^7ii3=  (6acc  +  263/K-^  +  . . . , 

U^^  =  2c2j"-s  +  (4c'aj  +  6d'2/)2^"-<^  +  •  •  •  , 

^123  =  %{^  -  ^y-'  +  (ri  -  3)  (26a;  +  2cy)z^-^  +  • .  • , 

U^  =  (n  -  3)  (2caj  +  6dy)z--^  +  •  •  • , 

^133=  (7i^2)(n~3Ky2^"-*  +  ..., 

^7^3,=  (n  ~  2)  (n  -  S)a^xz--^  +  •  •  • , 

Z7222  =  6c?2:«-3  +  (Qd'x  +  24e'3/K-^  +  •  •  • , 

C/;33=  a^(n-2)(n-3)(n~4)iC2/2;"-s  +  .... 

Evaluating  these  expressions  for  (0 ,  0 ,  1)  and  substituting  in  the 
determinant,  the  locus  of«  points  whose  first  polars  have  an  in- 
flexion at  (0,  0,  1)  is 


6ax  4-  2by  2bx-i-2Gy-\-aQ{n—2)z     a^{n  —  2)y 

2bx-\-2cy-\-aQ{n—2)z  2cx -\- My  aj^n  —  2)x 

S(^  —  2)2/  %{n  —  2)x  0 

or 

S{aci^  +  d'f)  —  {bv?y  +  cxy^)  —  aj^n  —  2)xyz  =  0  . 


=  0, 


16  THE  POLES  OF   A   RIGHT   LINE 

Thus  the  inflexion  cubic  has  a  double  point  at  (0 ,  0 ,  1)  with  the 
tangents  of  U,  x  =  0  and  3/  =  0 . 
For  a  cusp  U  takes  the  form 

a^a^z""-^  +  («a^  +  bxi'y  +  oxf  +  df)z''-^  +  . . .  =  0  . 

The  quantities  U^^^ ,  U-^j^^  ?  ^^^'f  ^^^  ^^®  same  as  before  except 
the  following : 

^113  =  2a^(7i  -  2>^  +  (n  -  3)  {6ax  +  2by)z'^  +  •  •  • , 
U,^  =  (n  -  3)  (26a^  +  2c2/>"-*  +  •  •  • , 

£^333  =  a^(n  -  2)  (n  -  3)  (n  -  4)x'z^-'  +  . . .  ; 

and  the  inflexion  cubic  is 

Qax  +  2by  +  2ao(^  -  2)^     26a;  +  2cy     2a^{n  —  2)x 


or 


2bx  +  2c2/  2cx  +  6(^y  0 

2ao(n  —2)x  0  0 

a;2(c£c  +  Sdy)  =  0  ; 


=  0, 


so  that  the  cubic  reduces  to   three  straight  lines,  two  of  which 
coincide  with  the  cuspidal  tangent. 

If  the  point  (x'j  y  ,  z) ,  whose  inflexion  cubic  is  considered,  is 
a  double  point  of  U ,  every  line  in  the  plane  intersects  the  cubic 
in  three  points  which  satisfy  xU^  +  yU^  -\-  zU^  =  0  -,  or  at  a 
double  point  of  U  three  poles  of  any  pencil  have  an  inflexion. 
If  the  line  pass  through  a  double  point  of  U ,  since,  as  we  have 
seen,  the  inflexion  cubic  has  there  a  double  point,  two  of  the 
three  intersections  are  represented  by  the  double  point.  The 
tangents  to  the  first  polar  of  the  double  point  meet  it  in  three 
points  which  are  coincident,  though  not  on  the  same  branch. 
They  thus  satisfy  the  algebraic  conditions  and  count  for  two  of 
the  tangents  required.  Corresponding  to  the  third  intersection 
is  an  inflexional  tangent,  strictly  speaking,  or  else  a  line  form- 


WITH   RESPECT   TO   A   CURVE   OF   ORDER   n.  17 

ing  part  of  a  degenerate  polar. ^  If  the  line  is  tangent  to  C/'  at 
the  double  point  its  three  intersections  with  the  inflexion  cubic 
coincide.  Two  of  these  as  before  are  at  the  double  point  and 
have  the  same  effect,  while  the  third,  which  is  consecutive  to 
the  double  point,  gives  the  line  itself  as  inflexional  tangent  to 
the  polar  of  a  point  in  it  and  at  the  point  itself. 

Passing  to  the  case  where  (x,  y' ,  z)  is  a  cusp  on  U,  any  line 
which  does  not  go  through  the  cusp  meets  the  inflexion  cubic  in 
two  coincident  points  on  the  cuspidal  tangent,  and  in  a  third 
point  on  the  other  right  line  which  makes  up  the  cubic.  The 
oint  in  the  cuspidal  tangent  is  one  whose  first  polar  has  a  dou  ble 
point  at  (x\  y' ,  z) ,  and  the  tangents  at  this  double  point  count  for 
two  of  the  required  lines.  The  third  point  is  the  one  whose  first 
polar  has  the  cuspidal  tangent  for  inflexion  tangent.  An  ordi- 
nary line  through  the  cusp  has  there  three  intersections  with  the 
inflexion  cubic.  The  cuspidal  tangent  counted  twice  corresponds 
to  two  of  the  intersections  as  cuspidal  tangent  to  their  first  polar 
curves,  and  as  inflexional  tangent  corresponds  to  the  third. 
Finally,  the  cuspidal  tangent  has  three  intersections  with  the  in- 
flexion cubic  at  the  cusp,  two  represented  by  the  cuspidal  tan- 
gent as  before,  and  the  third  giving  the  line  itself  as  inflexion 
tangent  to  the  polar  of  one  of  its  points  at  that  point. 

An  examination  of  the  quartic 

U=:  x^y^  +  yh^  +  ^7?  -f-  \xy^  —  lyzx^  —  ^zxf"  =  0 

will  serve  to  illustrate  these  different  cases.  No  simpler  example 
is  possible,  for  this  is  the  lowest  order  of  curve  for  which  the 
Hessian  and  the  Steinerian  are  distinct,  and  all  but  one  of  the 
poles  which  could  be  fixed  by  means  of  cusps  and  double  points 
are  at  the  vertices  of  the  triangle  of  reference.  The  cuspidal 
tangents  are  2^  —  ic  =  0 ,  3/  —  2  =  0 ;  and  those  at  the  double  point 
are  x  +  2y  =  0 ,  and  2a3  -h  2/  =  0 .     Differentiating 

^  This  third  intersection  agrees  with  Cremona's  statement,  Introduction,  §  10, 
No.  47,  that  one  curve  of  a  pencil  will  have  an  inflexion  at  a  double  pole, 
though  he  approaches  the  question  from  an  entirely  different  standpoint  and 
does  not  consider  whether  or  not  the  point  is  a  true  inflexion. 


18 


THE  POLES   OF  A   RIGHT  LINE 


U^  =  l3?y  4-  22/^2  4-  \x^  -  Izt?  -  4a;2/2! , 

U^  =  21/^2  +  2a^z  +  ^xyz  —  ^yx"  —  2xy^ ; 

U^^  =  22/^  +  2^2  -  Ayz,  £7-2,  =  2ar^  +  2^2  -  4a;2 , 

CTj^  =  Axy  4-  f  2^  — •  42;ic  —  4y2,        U^  =  41/2  +  52a;  —  Ix^  —  Axy, 

U,,=  4xz  +  6yz-4xy  -  2f,        ^^3=  V  -h  ^^  +  ^xy; 

^111=  ^222=  ^-  =  0 


'333 


Forming  the  partial  derivatives  of  the  third  order  and  evaluating 
for  the  points  (0,  0,  1),  (1,  0,  0),  etc., 


f^n2  =  42/— 42 
^113  =  43— 4y 
U,22  =  4x  —  4z 
J7i23  =  5z  —  4x- 
Z7,23  =  4z  — 4a; 
f^233  =  4y  +  5a; 
Z7i83  =  4a;  +  5y 


(0,0,1) 

(1,0,0) 

(1,1,0) 

—4 

0 

4 

4 

0 

—4 

—4 

4 

4 

5 

—4 

—8 

4 

—4 

—4 

0 

5 

9 

0 

4 

9 

(1,1,1) 

0 
0 
0 
—3 
0 
9 
9 


(^) 


=  0, 


The  inflexion  cubic  for  the  double  point  (0,  0,  1)  is 

4(2  —  y)         52  —  4(x  +  y)     4x  -\-  by 
bz  —  4(x  +  y)  4(2  —  cc)         5a;  +  4y 

4ic  +  5y  503  +  42/  0 

which  reduces  to 

16(j»3  +  2/^)  -  62(iK2  _|_  y2^  ^  38iC2/(i»  +  2/)  -  15»53/^  =  ^ ; 
and  for  the  cusp  (1 ,  0,  0) 

0  4(2/  -  z)  4(2  -  y) 

^{y  ~-  ^)  4(0;  —  2)  52;  -—  4(ic  +  y) 

4(2  —  2/)         ^^  —  4(a;  -\-  y)  4:X  +  by 

(y-.)^(y-2.)  =  0. 


(£) 


=  0, 


or 


WITH  EESPECT  TO  A  CURVE  OF  ORDER  n. 


19 


(A)  has  at  (0 ,  0 ,  1)  a  double  point  with  tangents  x  -\-  2y  =  0 
and  2x  -^  y  =  0;  and  at  the  two  cusps  the  inflexion  cubics  re- 
duce to  the  cuspidal  tangent  counted  twice  and  a  third  straight 
line  through  the  cusp,  according  to  the  general  theory. 

I.  When  the  line  has  a  pole  at  a  point  not  on  the  curve. 

Example  1.  The  point  (1,  1,  1)  has  the  inflexion  cubic 


m 


or 


0 

--Sz 

9z-Sy 

-Sz 

0 

9z  -Sx 

dz^Sy 

9z—3x 

dx+9y 

0, 


27(18»*  -  Syz"  -  Sxz^  +  2xyz)  =  0 , 


The  polar  of  a  point  on  any  line  will  have  one  inflexion  at 
(1,  1,  1)  if  there  is  a  common  intersection  of  X,  (D),  and  the 
first  polar  of  the  point  in  which  Z7j,  U^,  and  U^  have  been 
evaluated  for  (1 ,  1 ,  1) ;  and  there  will  be  three  simultaneous 
intersections  if  (1,  1,  1)  is  a  pole  of  the  line  selected.  The 
line  X  +  y  -{-  lOz  =  0  has  a  pole  at  (1 ,  1 ,  1)  and  its  three  inter- 
sections with  (Z))  are  (1 ,  —  1 ,  0) ,  (1 ,  — •  6 ,  J)  and  (1 ,  —  |^ , 
—  ■^^) .     The  respective  polars  of  these  points  are 

(x  —  y)  (^izx  -f  4zy  —  4xy  —  z^)  =  0 , 

(z  —  x)(2y^  +  19yz  -f  2Qxz  —  26xy)  =  0, 

(y  —  z)  {26xy  —  26yz  —  19xz  —  2x^)  =  0 ; 

showing  that  the  polars  are  degenerate,  and  the  straight  lines 
which  pass  through  the  pole  correspond  to  the  inflexional  tan- 
gents or  tangents  at  a  double  point  on  a  proper  curve. 

Example  2.  The  line  x  +  y  —  2z  =  0  has  a  pole  at  (1 ,  1 ,  0) 
for  which  the  inflexion  cubic  is 


(E) 


4y  —  4z  4x  -\-  4y  —  ^z     9z  —  4x  —  Sy 

4x+  4y  —  Sz         4x  —  4z  dz  —  Sx  —  4y 

9z  —  4x—  Sy    9^  —  8a;  —  4y         9x  +  9y 


0, 


20  THE   POLES   OF   A   RIGHT   LINE 

which  reduces  to 

4(x'  -\-f)-\-  S{x'y  +  xf)  +  6S{yz'  +  xz') 

-  2S(yh  +  xh)  -  4.0xyz  -  54»»  =  0 . 

The  intersections  of  the  line  with  the  cubic  (E)  are 

a;=l,l,  cd;  y=l,l,  oo;  2;=!,  1,1; 

and  the  line  is  tangent  to  the  cubic  at  (1 ,  1 ,  1) ,  while  the  third 
intersection  is  at  infinity.     The  polar  of  (1 ,  1 ,  1)  is 

z{Syz  +  Sxz  —  2xy)  =  0 , 

a  degenerate  cubic  which  is  to  be  counted  twice  on  account  of 
the  condition  of  tangency. 

II.  Every  line  has  a  single  pole  at  the  double  point.  (See 
§5,1.) 

Example  3.  The  line  176aj+  14%  +  15^;==  0  has  intersec- 
tions with  (A)  at  (6,  -  9,  19),  (45,  -  60,  68),  and  (-  35,  20, 
212),  with  corresponding  polars 

dQx'z  -  56x^y  -  26xy^  +  2Qyh  +  lOlxyz  -  Zyz^  -  ^-xz''  =  0 , 
A.^zf  -  46ici/2  ^  266x''y  +  256 A 

-f  400xyz  -  60xz^  -  ^fyz^  =  0 , 
4cdiyh  —  494:xy^  —  20yz^  —  ^f-yz^ 

+  1120xyz  -  384a;2^  +  384aj22j  =  0  . 

Changing  to  Cartesian  coordinates,  and  making,  for  each  curve, 
the  tangent  at  the  origin  the  3^-axis,  these  equations  reduce  to 

33(12^2  _  iQ^^  ^  26a;  +  278^  -  21)  +  40f  =  0 , 

x(42y  —  4xy—  ISy'  +  4x  —  ^^^-)  +  Idy^  =  0 , 

x(66bf  -  S52y  -  10)  +  39713/*  =  0 . 

This  shows  that  there  exist  three  real  points  in  the  Jine  176ic 
+  1492/4-  15^;=  0  whose  first  polars  have  an  inflexion  in  the 
strict  sense  of  the  word  at  the  pole  of  the  line  (0 ,  0 ,  1)  . 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  71.        21 

Example  4.  The  line  x  —  2y  =  0  passes  through  the  double 
point  and  its  intersections  with  (J.)  are 

'K  =  0,0,^f;  2/  =  0,0,5«j;  ^=1,1,1. 

The  double  point  counted  twice  has  corresponding  to  it  the  tan- 
gents to  its  first  polar  at  (0 ,  0 , 1)  ;  and  the  polar  of  (10 ,  5 ,  31)  is 

42  {y^z  -  xy^)  +  52  {xh  -  x^y)  +  ^bxyz  +  Zbyz^  +  %6  rf  =  o . 

This  equation  transformed  as  before  gives 

X  (65  -  ^xy  -  34?/  +  8ic  +  1402/")  ~  Z^2f  =  0, 

which  shows  an  inflexion  at  (0,  0,  1). 

Example  5 .  The  line  2=0  bears  no  special  relation  to  the  in- 
flexion cubic  for  (0,  0,  1),  but  has  a  pole  at  that  point,  and  by 
choosing  suitable  coordinates  any  line  may  be  taken  for  s  =  0 . 
We  may  then  find  the  points  whose  first  polars  have  an  inflexion 
at  (0,  0,  1)  by  the  following  general  method. 

Let  cc  —  %  =  0  be  the  equation  of  the  line  joining  (0 ,  0,  1) 
to  an  intersection  of  (A)  with  i  =  0 ;  then  the  tangent  to  the 
first  polar  of  the  point  is  cc  -f  %  =  0  (see  §  5,  I) ,  and  h  may 
be  so  determined  that  a;  +  %  =  0  is  tangent  at  an  inflexion. 
The  polar  of  (^ ,  1 ,  0)  is 

-f-  I  jcz^  —  2sic^  —  Axyz  =  0  ; 
and  the  tangent  to  T^^  at  (0 ,  0 ,  1)  is 

x(U+5)-i-y{5k+  4)  =  0. 

This  must  meet  F^  in  three  points  at  (0,  0,  1).     Substituting 

/5Z;-f4\         ,,     . 
^  ^  ""  ^  V  4F+5  j^  ^^^  ^^^^^"^^  ^  "=  ^^ 

^^1     4;^+ 5      J 

^    r  10k' -h  isk -{- s     ,     /5^  +  4\n     ^ 


22  THE  POLES   OF   A   RIGHT   LINE 

The  coefficient  of  y  vanishes  identically  since  the  line  is  tangent 
hy  hypothesis.  For  three  values  of  y  to  be  equal  to  zero  the 
coefficient  of  3/^  must  vanish,  and 

()^+l)(8y^2+ll^+8)  =  0, 

which  gives  one  real  and  two  imaginary  values  for  h .  When 
A;  =  —  1  the  polar  of  (  —  1 ,  1 ,  0)  is 

{x  —  y)  (z^  +  4xy  —  4xz  —  4yz)  =  0 , 

another  degenerate  polar;  while  the  two  remaining  points  in 
2=0  whose  first  polars  have  an  inflexion  at  (0,  0,  1)  are  im- 
aginary.^ 

III.  Every  line  has  a  double  pole  at  the  cusp  (see  §  5). 

Example  6.  Intersecting  {B)  by  the  line  a;  =  0,  we  obtain 
(0,  1,  1)  twice,  and  (0,  2,  1).  The  tangent  to  the  polar  of 
(0,  1 ,  1)  is  indeterminate,  and  this  point  corresponds  to  the  polar 
which  has  a  double  point  at  the  pole.     The  polar  of  (0,  2,  1)  is 

2x^y  —  ^xy"^  —  Sxyz  —  2xh  +  4yz^  +  2yh  +  bxz^  —  0, 

and  its  tangent  at  (1 ,  0,  0)  is  2/  —  sj  =  0,  the  cuspidal  tangent. 
Transforming  so  that  y  —  z  —  0,  becomes  2  =  0  we  have 

<2  ~  72/ +  52;  -  10y2  +  42/2^)  +  62/^  =  0  ; 

and  2/  —  2J  =  0  is  therefore  tangent  at  an  inflexion. 

Cases  where  the  line  passes  through  a  cusp  or  coincides  with 
the  tangent  there,  and  other  special  positions  of  the  line  may 
readily  be  examined  by  similar  methods. 

1  Writing  the  polar  conic 

2a;2  4-2%2  — y2(5^  +  4)— 2x(4^  +  5)+4a;y(yfc  +  l)=:0, 

dividing  by  the  tangent 

x(4^  +  5) +2/(5^  +  4)^0, 

and  equating  the  coefl&cients  in  the  remainder  to  zero,  leads  to  the  same  values 
of  k. 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  n.        23 

§5. 

Poles  when  the  Base  Curve  has  Double  Points 
AND  Cusps. 

A  double  point  on  Z7is  a  pole  for  every  line  in  the  plane  and 
presents  several  peculiar  characteristics.  It  is  a  double  point 
on  its  own  first  polar  curve,  and  therefore  corresponds  to  itself 
as  a  point  on  the  Hessian  and  the  Steinerian.  It  is  a  double 
point  on  the  Hessian  ^  which  represents  always  two  of  the  points 
which  can  be  double  points  for  the  pencil  of  curves  belonging 
to  a  line  through  it,  and  is  therefore  a  double  point  on  the 
Steinerian.  This  is  a  particular  case  of  the  following  theorem 
proved  by  Henrici  ^  by  a  very  elegant  analysis  :  "  A  point 
whose  first  polar  has  a  cusp  is  a  cusp  on  the  Steinefian,  and  one 
whose  first  polar  has  two  double  points  is  a  double  point  on  the 
Steinerian."  ^  The  tangents  to  the  Hessian  at  the  double  point 
are  the  same  ^  as  those  of  U,  and  also  are  tangents  to  the  Stein- 
erian, since  they  are  line  polars  of  the  corresponding  point  on 
the  Hessian. 

This  pole  lies  on  Uy  the  Hessian  and  the  Steinerian  simulta- 
neously, irrespective  of  any  condition  introduced  by  the  position 
of  the  line  itself,  and  it  is  in  general  a  single  pole,  contrary  to 
the  usual  character  of  a  pole  on  the  Hessian  ;  but  the  Hessian 
includes  all  points  which  are  simply  double  points  on  U  as  well 
as  those  where  all  first  polars  touch.  A  cusp  is  a  double  pole 
for  all  lines  in  the  plane  since  all  first  polars  touch  there.  Thus 
there  is  a  lower  limit  for  the  number  of  single  and  double  poles 
found  on  U,  for  any  line  in  the  plane,  depending  upon  the  num- 
ber of  double  points  and  cusps  which  U  has.  One  of  these 
single  poles  may  change  into  a  double  pole,  and  a  double  pole 
into  one  of  higher  order,  for  certain  lines  in  the  plane — namely, 
lines  through  the  singular  points  and  tangents  at  those  points. 
This  lowest  number  is  increased  by  the  points  of  tangency  of  the 

*  Salmon,  Higher  Plane  Curves^  p.  60, 

2Proc.  L&nd.  Math.  Soc,  Vol.  II,  p.  112. 

^Cf.  Cremona,  Introductwn,  §  20,  No.  120.  If  a  first  polar  has  two  double 
points  p  and  j/,  the  pole  o  is  a  double  point  on  the  Steinerian  and  the  tangents 
at  0  are  line  polars  of  p  and  p''. 


24  THE   POLES   OF  A   RIGHT  LINE 

line  with  U,  Any  pole  which  is  common  to  i,  C/,  and  the 
Hessian  may  be  either  a  double  point  or  an  inflexion  :  if  it  is  on 
the  Steinerian  as  well  it  must  be  a  double  point. 

I.  When  L  does  not  pass  through  the  double  point  the  pole  is 
single,  and  the  tangent  to  the  first  polar  of  a  point  in  L  is  the 
harmonic  conjugate,  with  respect  to  the  tangents  of  U  at  the 
double  point,  to  the  line  joining  the  point  whose  polar  is  taken 
to  the  double  point.^ 

£7"  may  be  written 


a, 


rcyz""-^  +  (aa^  +  h7?y  +  cxy^  -f  dif)z'^^  -j-  . . .  =  0 


Let  L  be  the  line  2J  =  0  for  simplicity.  Then  any  point  {x^ ,  3/1 ,  0) 
on  L  has  for  polar 

The  tangent  to  Fj  at  (0 ,  0 ,  1)  is  xy^  +  yx^  =  0  ;  and  the  line 
through  (cCj,  y^,  0)  and  (0,  0,  1)  is  xy^  —  yx^  =  Oy  the  harmonic 
conjugate  of  the  tangent. 

In  particular  the  point  where  the  line  intersects  one  of  the 
tangents  at  the  double  point  has  a  polar  which  touches  the  other 
tangent  at  the  double  point. 

II.  When  L  passes  through  the  double  point  it  is  evident  that 
the  harmonic  conjugates  reduce  to  a  single  line  conjugate  to  X 
which  is  the  common  tangent  to  all  first  polars,  and  there  are 
two  coincident  poles.  This  is  one  of  the  special  cases  referred 
to  in  §3  (c),  for  X  passes  through  a  double  point  on  the  Stein- 
erian and  has  a  pole  on  the  Hessian.  The  two  points  compos- 
ing the  double  pole  must  be  regarded  as  lying  on  the  conjugate 
to  X,  while  L  simply  intersects  the  polars  at  the  double  point, 
having  two  points  in  common  only  with  the  polar  of  the  double 
point,  which  has  a  double  point  at  the  pole.  It  should  be  noted 
that  we  have  here  another  condition  for  a  double  pole. 

III.  If  L  is  tangent  to  C/"  at  a  double  point,  it  is  tangent  to 
all  first  polars  of  its  points,  and  to  the  Hessian  and  the  Stein- 
erian, and  has  three  consecutive  points  in  common  with  U,  the 

1  Cf.  Cremona's  theorem,  Introduction,  ?  13,  No.  74,  based  on  his  geometrical 
proof. 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  71.  25 

Hessian,  and  the  Steinerian.  The  three  intersections  with  the 
Hessian  count  for  three  double  points  of  the  pencil,  and  the 
double  pole  lies  on  the  line  itself. 

IV.  When  the  double  point  is  a  cusp,  the  two  tangents  coin- 
cide and  become  the  common  tangent  to  U  and  to  all  first  polars. 
This  is  also  a  condition  for  a  double  pole.  The  point  where  any 
line  intersects  the  cuspidal  tangent  is  the  one  whose  first  polar 
has  a  double  point,  counting  for  two  double  points  of  the  pen- 
cil as  in  the  ordinary  case  where  polar  curves  have  a  common 
tangent. 

Y.  Any  line  through  the  cusp  has  three  points  in  common 
with  the  Hessian,  since  a  cusp  on  U  is  a  triple  point  on  the 
Hessian  ^  consisting  of  a  cusp  with  a  simple  branch  through  it. 
This  pole  then  counts  for  three  double  points  of  the  pencil  and 
the  polar  of  the  cusp  itself  has  a  cusp  there.^  Any  line 
through  the  cusp  has  at  least  two  poles  on  the  cuspidal  tangent 
at  the  point  of  tangency,  but  this  double  pole  as  in  a  former  case 
cannot  be  regarded  as  lying  on  the  line.  The  pencil  at  a  cusp 
diifers  from  a  pencil  at  an  ordinary  point,  when  the  line  passes 
through  it,  only  in  having  a  cusp  at  the  pole  instead  of  a 
double  point.  The  cusp  on  U  should  be  a  cusp  on  the  Steinerian 
by  Henrici's  theorem  already  quoted  in  this  section,  ^  but  this 
is  a  special  case  since  the  cuspidal  tangent  forms  a  part  of  the 
Steinerian. 

VI.  If  L  is  tangent  to  CT"  at  a  cusp,  every  point  in  it  is  one 
whose  first  polar  curve  has  a  double  point  at  the  cusp.^  All 
first  polars  have  four  intersections  at  the  cusp,  making  a  pole  of 
order  4.  The  pairs  of  tangents  to  the  polars,  or  their  polar 
conies,  form  a  quadratic  involution  with  the  vertex  at  the  cusp 
in  which  one  of  the  two  double  elements  is  the  cuspidal  tangent 
itself. 

1  Salmon,  Higher  Plane  Curves,  p.  61. 

'  Cf.  Cremona,  §  14,  88b  :  If  a  base  point  of  a  pencil  is  a  cusp  for  one  it 
counts  for  three  double  points. 

'Also  compare  Cremona,  Introduction,  ^  20,  No.  121  :  If  a  first  polar  have 
a  cusp  p,  the  pole  is  a  cusp  on  the  Steinerian  and  has  the  line  polar  of  p  for 
cuspidal  tangent. 

*  Cremona,  Introduction^  §14,  88d. 


26  THE  POLES  OF  A   RIGHT  LINE 

Let 

^1  =  ^1^1  +  2/1^2  +  ^1^3=0,'^ 

^2=^2^1  +  ^2^2+^2^3=0, 

be  the  first  polars  of  any  two  points  on  the  cuspidal  tangent. 
The  pairs  of  tangents  at  the  cusp  (x',  y' ,  z)  are  given  by 

+ yi  TJ'm  +  h  U^^,^\  +  22/2  {x,  U\^  +  y,  U\,,  +  z,  U\^-\  +  2zx  [x^  U\,, 
+  2/1  U\,s  +  h  U^^l  +  2xy  [x,  U\n  +  Vx  U'n,  +  ^i  U\,,-\  =  0 , 

and  a  similar  expression  C^ .  Any  other  point  on  the  cuspidal 
tangent  is  (a?j  +  Xx^,  y^  +  ^y^y  z^  -\-  Xz^),  and  the  tangents  to  its  first 
polar  are  given  by  C^  +  W^^O.  Two  curves  of  the  system 
will  have  a  cusp  corresponding  to  the  double  elements  of  the 
involution.  These  are  given  by  the  values  of  X  which  make 
Cj  +  ^Cg  a  perfect  square ;  and,  since  one  of  the  pairs  is  the  cus- 
pidal tangent  and  therefore  real,  the  involution  is  hyperbolic  or 
non-overlapping.^ 

§6. 

Intersections  of  Higher  Order  with  the  Steinerian. 

The  character  of  the  poles  conditioned  by  the  line  having 
ordinary  contact  with  the  Steinerian  has  already  been  discussed 
in  §3,  and  it  was  shown  that  parallel  theorems  for  the  Hessian 
cannot  be  deduced.  The  condition  of  passing  through  a  double 
point  dn  the  Steinerian  is  not  equivalent  to  the  condition  of  tan- 
gency,  but  a  line  may  pass  through  a  double  point  on  the  Steiner- 
ian and  have  no  pole  at  the  corresponding  point  on  the  Hessian. 
A  line  through  a  double  point  on  U  however,  since  it  is  a  dou- 
ble point  on  both  these  loci,  has  a  pole  on  each,  the  double  point 
itself.  This  is  a  special  case  depending  on  the  relation  of  the 
line  to  the  base  curve. 

When  the  line  has  three  intersections  with  the  Steinerian  at 
any  point  the  following  distinctions  arise : 

I.  The  three  points  may  be  consecutive  on  the  same  branch 
and  the  line  is  tangent  at  a  point  of  inflexion. 

^  Scott,  Analytical  Oeomdry^  p.  162. 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  71.  27 

Regarding  this  as  the  limiting  position  of  two  points  of  tan- 
gency  with  the  Steinerian  which  unite  to  form  an  inflexion,  the 
corresponding  points  on  the  Hessian  unite,  while  the  common 
tangents  to  the  polar  curves  move  to  coincidence  in  the  same 
way  that  the  points  of  tangency  on  the  Steinerian  do,  giving 
three  consecutive  points  in  the  same  straight  line  for  all  polars.^ 
The  common  tangent  is  an  ordinary  tangent  to  the  Hessian,  since 
only  two  points  have  moved  to  coincidence ;  and  the  line  has 
here  a  pole  of  order  3 ,  which  counts  for  three  double  points  of 
the  pencil. 

II.  Two  points  may  be  consecutive  and  the  third  in  another 
branch,  so  that  the  line  is  tangent  to  the  Steinerian  at  a  double 
point. 

The  condition  of  tangency  gives  a  pole  on  the  Hessian  of  order 
2 .  The  polar  of  the  point  has  three  double  points,  two  of  which 
coincide  at  the  pole  as  in  the  ordinary  case  but  still  count  for 
two. 

III.  If  the  Steinerian  admits  a  triple  point,  the  three  inter- 
sections may  be  on  three  different  branches. 

Corresponding  to  a  triple  point  with  three  simple  branches  is 
a  polar  curve  with  three  double  points,  but  no  condition  is 
necessarily  imposed  on  any  pole.  A  line  tangent  to  the  Stein- 
erian at  the  triple  point  meets  it  in  four  points  there,  and  from 
the  condition  of  tangency  the  corresponding  point  on  the  Hessian 
is  a  pole  of  order  2 ,  at  which  the  double  point  counts  for  two. 

A  triple  point  on  the  Steinerian  formed  by  a  simple  branch 
through  a  cusp,  has  a  polar  with  two  double  points  and  a  cusp. 
No  pole  is  defined  for  a  line  passing  through  the  point  by  this 
condition.  If,  however,  the  line  is  tangent  to  the  simple  branch 
at  the  double  point,  the  corresponding  point  on  the  Hessian  is  a 
pole  of  order  2 .  The  polar  of  the  triple  point  has  a  double 
point  at  the  pole,  counting  for  two,  and  a  cusp  elsewhere.  If 
the  line  is  a  cuspidal  tangent,  the  cusp  is  at  the  pole  and  counts 
for  three  double  points ;  the  double  point  elsewhere  counts  for  a 
single  double  point  of  the  pencil. 

Any  multiple  point  which  the  Steinerian  may  have  with  dis- 

*  Cf.  Cremona,  Introduction,  §  20,  No.  119. 


28  THE   POLES    OF    A   RIGHT   LINE 

tinct  or  coincident  tangents  to  the  several  branches  will  give 
analogous  results.  '      • 

§7. 

The  Base   Curve  with   Triple   Points  and   Multiple 
Points  of  Higher  Orders. 

The  first  polars  of  any  two  points  in  the  plane  pass  through  a 
triple  point  on  TJ  twice,  and  the  four  intersections  compose  a 
pole  of  order  4  for  any  line  in  the  plane.  The  polar  of  the 
triple  point  has  there  a  triple  point  with  the  tangents  of  IJ  for 
its  three  tangents.  A  pencil  of  first  polars  has  ordinarily 
3(n  —  2)^  curves  which  may  have  a  double  point,  and  these  cor- 
respond to  the  intersections  of  the  line  with  the  Steinerian ;  but 
in  the  case  considered  the  first  polar  of  every  point  in  the  plane 
has  a  double  point  at  the  triple  point  of  TJ,  Thus  the  ordinary 
Steinerian  is  indeterminate  when  the  base  curve  has  a  triple 
point,  but  there  is  only  a  finite  number  of  points  whose  second 
polars  have  a  double  point,  and  the  locus  for  these  can  be  found, 
at  least  theoretically.  Adopting  the  notation  suggested  by 
Salmon,^  this  locus  is  the  second-Steinerian  of  order  6(71  —  3)^; 
and  the  corresponding  second-Hessian  is  the  locus  of  such  double 
points,  among  which  are  the  triple  points  of  Z7,  and  is  of  order 
12(n  —  3) .  In  general  the  i^-Steinerian  and  ??-Hessian  are  of 
orders  Zd-{n  —  i^—\f  and  Z&\n  —  ??  —  1)  respectively. 

It  has  been  shown  that  the  inflexion  cubic  for  a  cusp  is  com- 
posed of  the  cuspidal  tangent  taken  twice  and  another  line 
through  the  cusp  ;  and  it,  as  well  as  the  Steinerian,  is  therefore 
degenerate  when  U  has  a  cusp.  If  U  has  a  triple  point,  the  in- 
flexion cubic  for  that  point  is  indeterminate,  as  may  readily  be 
proved  by  evaluating  the  determinant  for  such  a  point.  It  is 
evident  that  the  Steinerian  is  connected  with  the  inflexion  cubic 
for  any  point  as  the  Hessian  is  with  the  inflexion  locus  for 
any  line. 

At  a  multiple  point  of  order  h,  first  polar  curves  will  have  a 
multiple  point  of  order  ^  —  1  in  general,  but  the  polar  of  the 

'  Higher  Piane  Curves,  p.  365. 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  71.        29 

multiple  point  itself  will  have  precisely  the  same  multiple  point 
that  U  has.  The  number  of  poles  at  the  point  may  be  increased, 
as  we  have  seen,  by  the  position  of  the  line  or  by  coincidence  of 
tangents. 

There  are  certain  harmonic  relations  which  govern  the  tangents 
to  first  polar  curves  at  a  multiple  point  of  U  in  consequence  of 
the  harmonic  properties  of  poles  and  polars,  and  we  shall  con- 
clude this  paper  with  an  outline  of  these  relations. 

A,     Triple  Points, 
I.    CT'has  an  ordinary  triple  point  at  (0,0,1)  and  is  of  the 
form 

a^fcy(x  —  y)z'^^  +  (ax^  +  boc^y  -f  cx^f  +  dxi^  -f-  ey*)a"-*  +  •  •  •  =  0 . 

(a)  L  does  not  pass  through  the  triple  point. 
The  polar  of  any  point  {x^ ,  3/^ ,  2J  J  in  X  is 

X,  \_a^{2x  -  y)z--'  +..-]+  2/1  [%^{^  -  2^^-'  +  •  •  •  ] 

+  3  Ja,(n  -  3)a32/(aJ  -  yK"' +•••]  =  ^  > 

with  tangents  at  (0,  0,  1)  given  by 

^i3/(2aJ  -  y)  +  yA^  -  22/)  =  0 . 

There  are  two  double  rays  in  this  quadratic  involution  and  there- 
fore two  polar  curves  have  a  cusp  at  the  triple  point.  The  two 
factors  of  the  first  term  are  y  and  the  harmonic  conjugate  of  y 
with  respect  to  the  other  two  tangents ;  and  the  second  term  is 
the  corresponding  expression  for  x .  This  relation  of  tangents  is 
similar  to  the  case  for  a  double  point. 

(6)  L  passes  through  the  triple  point,  or  L  =  x  -~ky=0. 

In  this  case  the  tangents  are  contained  in 

ky(2x--y)  +  x{x-2y)  =  0, 

and  are  the  same  for  all  curves  of  the  pencil,  except  for  (0 ,  0 ,  1) 
which  has  the  tangents  of  U?    This  again  is  an  extension  of  the 

1  Cf.  Cremona,  Introduction,  §  10,  No.  48  :  If  ^  and  A'  are  tangent  to  all 
curves  of  a  pencil  at  a,  a  curve  may  be  found  which  has  a  for  a  triple  point. 
If  A  and  A^  coincide  all  curves  have  a  cusp. 


30  THE   POLES   OF   A   RIGHT   LINE 

case  for  a  double  point,  where  all  polar  curves  touch  the  har- 
monic conjugate  of  the  line  with  respect  to  the  two  tangents  of 
Uat  the  double  point. 

(c)  L  is  tangent  to  U  at  the  triple  point. 

Let  Lhe  x  —  y  =  0  y  and  the  tangents  are  x^  —  y^  =0,  Thus 
all  polars  have  two  common  tangents,  the  line  itself  and  its 
harmonic  conjugate  with  respect  to  the  other  two.  * 

II.  U  has  a  cuspidal  branch  at  (0 ,  0 ,  1),  or  its  lowest  term  is 
a^x^yz"*-^ .     The  tangents  to  first  polars  are  given  by 

x(2x^y  +  y^x)  =  0  . 

(a)  A  line  not  passing  through  the  triple  point  will  have  for 
the  pairs  of  tangents  to  its  first  polar  curves  the  cuspidal  tan- 
gent with  a  pencil  of  lines  through  the  triple  point.  One  polar 
of  the  pencil  with  have  a  cusp  there. 

(6)  A  line  i  =  a?  —  %  =  0  through  the  triple  point  has  tan- 
gents given  by  x{2ky  -f  ic)  =  0 ,  and  they  are  the  same  *  for  the 
polars  of  all  its  points.  The  polar  of  the  triple  point  has  a 
triple  point  and  only  this  one  polar  of  the  pencil  has  a  cusp. 

(c)  L  is  tangent  at  the  triple  point. 

Let  L  =  x—  0 f  the  cuspidal  tangent,  and  the  tangents  to 
first  polars  are  given  by  cc^  =  0 ;  and  all  first  polars  of  the 
pencil  have  a  cusp  at  the  triple  point. 

On  the  other  hand  if  i  =  y  =  0 ,  the  tangents  are  xy  =  0 , 
and  only  the  polar  of  the  triple  point  has  a  cusp. 

B.     Quadruple  Points. 

I.  U  has  a  quadruple  point  with  four  ordinary  branches 
through  it,  or 

Z7=  xy  {x  —  y)(x-{-  hy)  2"-^  +  {a:i(^  +  •  •  •)  s""^  +  . . .  =  0  . 

The  polar  of  {x^,  y^,  z^  has  tangents  at  (0,  0,  1)  given  by 
X,  [Zx'y  +  2  (^  -  1)  xy"-  -  hi^-] 

+  y,\^  +2{k^  l)xy'^Skxy']  =  0, 
and  the  four  double  elements  of  the  involution  are  the  tangents 


WITH  RESPECT  TO  A  CURVE  OF  ORDER  71.        31 

to  first  polars  which  have  a  cuspidal  branch  at  (0,  0,  1).  As  in 
the  preceding  cases  the  line  x  —  ¥y  =  0  has  the  same  tangents 
for  all  polars  except  that  of  the  quadruple  point.  If  the  line  is 
the  tangent  ic  —  y  =  0,  we  have 

{x  -  y)[f  +2{h+l)xy^  ky^  =  0, 

which  may  be  written 


{x  -  y)  \x  (a;  -f  2^  +  \y)  ^xy+y{2x  +  ley)]  =  0  ; 


where  x  -\-  2k  -^  ly  is  the  harmonic  conjugate  ofx  —  y  with  re- 
spect to  y  and  x  -\-  ky,  and  2x  -f  ky  is  the  harmonic  conjugate 
of  X  —  y  with  respect  to  x  and  x  +  ky.  This  may  be  written 
in  other  forms, 


{x  -  y)  [x  {x  +  2k+  ly) 

—  x{x  +  ky)  +  {x-Jrky){x  +  yj]  =  0,  etc., 

showing  other  combinations  of  harmonic  conjugates. 

II.  One  of  the  branches  is  cuspidal,  and  C/^may  be  written 

x\x  —  y){x  +  kyy-^^  -\ =  0 .     The  polar  curve  has  tangents 

given  by 

x^  \\x^  +  3(7^  ^\Yy-  2kxy^  +  y^  [{k  -  1^  -  2k  ^xy]  =  0 . 

For  any  line  x  —  k'y  =  0  the  cuspidal  tangent  is  a  common 
tangent  to  all  first  polars  as  before.  For  the  tangent  x  —  y  —  O 
we  obtain 

a;  [(3  +  ^  )a;'  -  (5ifc  -l)xy  -  2kf]  =  0 ; 

and  for  the  cuspidal  tangent,  ic  =  0 , 

x\k{x  —  2/)  —  (a;  +  %)]  =  0, 

giving  combinations  of  harmonic  conjugates. 

III.  Two  of  the  branches  are  cuspidal  and  U  has  for  its 
lowest  term  x^yh'^~'^ .     The  tangents  for  any  polar  are 

^y{^iy  +  Vi^)  =  0 , 

and  any  polar  has  a  triple  point  with  two  fixed  tangents,  while 


32  THE   POLES   OF   A   KIGHT   LINE 

the  third  belongs  to  a  pencil  through  the  point.  If  L  =  x  —  ky 
=  0 ,  the  tangents  are  all  the  same  ;  while  for  one  of  the  tangents 
to  C7',  aj  =  0 ,  we  have  ar^y  =  0 . 

C.   Quintuple  Points, 
I.  The  five  branches  are  simple,  and  U  has  for  its  lowest  term 
xy(x  —  y){x-{-  ky)  {x  +  ly)z'^^.     The  tangents  to  first  polars  are 
given  by 

jCj  [4ar'y  +  3(Z  +  ^  -  l^y^  -2{k  +  l-^  kl)xf  -%*] 

+  Vil^*  -{■2(l-\-k^  lyy  ^2.{k-\-l-  kiyf  -  4%^]  =  0 , 

and  the  six  double  elements  of  the  involution  are  the  cuspidal 
tangents  to  first  polars  at  the  multiple  point.  For  L^x  —  k'y 
==  0  the  tangents  are  the  same  for  all  first  polars.  The  tangents 
to  first  polars  of  the  pencil  belonging  to  a;  —-  3/  =  0  are 

{x  —  y)  \xy{l  +  ^  +  2a;  +  2kl  +  l-[-  ky) 

+  (a;  4-  y)  (aJ  +  %)  i?^  +  ly)]  =  0, 

where  the  first  term  in  the  bracket  is  composed  of  the  product 
of  two  tangents,  x  and  y ,  and  the  harmonic  conjugate  of  a;  —  3/ 
with  respect  to  aj  +  %  and  x  -\-  ly ;  and  the  second  term  is  the 
product  of  X  -{-  ky  and  x  +  ly,  and  the  harmonic  conjugate  of 
X  —  y  with  respect  to  x  and  y .  By  this  selection  the  result  is 
more  simple  on  account  of  the  symmetry  of  the  pairs  of  tangents. 
Arranging  the  pairs  in  difierent  order  a  third  term  appears ;  for 
instance 


(x  -  y)[x{x  +  ly)  (a?  +  2^  -f  ly) 

+  y{x  +  ky)  {x  +  ly)  +  lxy{x  +  %)]  =  0 . 

Combinations  of  cuspidal  branches  give  results  corresponding 
to  those  obtained  for  lower  orders  of  multiplicity,  and  these 
methods  may  be  extended  to  higher  orders  where  the  branches 
are  simple  and  cuspidal.  The  equations  involving  the  tangents 
will  allow  various  combinations  of  harmonic  conjugates,  and  it 
is  to  be  observed  that  the  forms  can  be  made  more  symmetric 
when  the  multiple  point  on  U  is  of  odd  order. 


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PAT.  JAN.  21,  1908 


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